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If the option value deviates from the value of the replicating portfolio, investors can create an arbitrage positionThe Determinants of Value The binomial model provides insight into the determinants of option value. The value of an option is not determined by the expected price of the asset but by its current price, which, of course, reflects expectations about the future. This is a direct consequence of arbitrage. If the option value deviates from the value of the replicating portfolio, investors can create an arbitrage position, i.e., one that requires no investment, involves no risk, and delivers positive returns. To illustrate, if the portfolio that replicates the call costs more than the call does in the market, an investor could buy the call, sell the replicating portfolio and guarantee the difference as a profit. The cash flows on the two positions will offset each other, leading to no cash flows in subsequent periods. The option value also increases as the time to expiration is extended, as the price movements (u and d) increase, and with increases in the interest rate. While the binomial model provides an intuitive feel for the determinants of option value, it requires a large number of inputs, in terms of expected future prices at each node. As we make time periods shorter in the binomial model, we can make one of two assumptions about asset prices. We can assume that price changes become smaller as periods get shorter; this leads to price changes becoming infinitesimally small as time periods approach zero, leading to a continuous price process. Alternatively, we can assume that price changes stay large even as the period gets shorter; this leads to a jump price process, where prices can jump in any period. In this section, we consider the option pricing models that emerge with each of these assumptions. The Black-Scholes Model When the price process is continuous, i.e. price changes becomes smaller as time periods get shorter, the binomial model for pricing options converges on the BlackScholes model. The model, named after its co-creators, Fischer Black and Myron Scholes, allows us to estimate the value of any option using a small number of inputs and has been shown to be remarkably robust in valuing many listed options. The Model While the derivation of the Black-Scholes model is far too complicated to present here, it is also based upon the idea of creating a portfolio of the underlying asset and the riskless asset with the same cashflows and hence the same cost as the option being valued. The value of a call option in the Black-Scholes model can be written as a function of the five variables: Note that e-rt is the present value factor and reflects the fact that the exercise price on the call option does not have to be paid until expiration. N(d1) and N(d2) are probabilities, estimated by using a cumulative standardized normal distribution and the values of d 1 and d2 obtained for an option. The cumulative distribution is shown in Figure 5.4: In approximate terms, these probabilities yield the likelihood that an option will generate positive cash flows for its owner at exercise, i.e., when SK in the case of a call option and when KS in the case of a put option. The portfolio that replicates the call option is created by buying N(d1) units of the underlying asset, and borrowing Ke-rtN(d2). The portfolio will have the same cash flows as the call option and thus the same value as the option. N(d1), which is the number of units of the underlying asset that are needed to create the replicating portfolio, is called the option delta . A note on estimating the inputs to the Black-Scholes model The Black-Scholes model requires inputs that are consistent on time measurement. There are two places where this affects estimates. The first relates to the fact that the model works in continuous time, rather than discrete time. That is why we use the continuous time version of present value (exp-rt ) rather than the discrete version ((1+r)-t). It also means that the inputs such as the riskless rate have to be modified to make the continuous time inputs. For instance, if the one-year treasury bond rate is 6.2%, the riskfree rate that is used in the Black Scholes model should be The second relates to the period over which the inputs are estimated. For instance, the rate above is an annual rate. The variance that is entered into the model also has to be an annualized variance. The variance, estimated from ln(asset prices), can be annualized easily because variances are linear in time if the serial correlation is zero. Thus, if monthly (weekly) prices are used to estimate variance, the variance is annualized by multiplying by twelve (fifty two). 5.2: Valuing an option using the Black-Scholes Model On March 6, 2001, Cisco Systems was trading at $13.62. We will attempt to value a July 2001 call option with a strike price of $15, trading on the CBOT on the same day for $2.00. The following are the other parameters of the options: * The annualized standard deviation in Cisco Systems stock price over the previous year was 81.00%. This standard deviation is estimated using weekly stock prices over the year and the resulting number was annualized as follows: Weekly standard deviation = 1.556% Annualized standard deviation =1.556%*52 = 81% * The annualized treasury bill rate corresponding to this option life is 4.63%. The inputs for the Black-Scholes model are as follows: Current Stock Price (S) = $13.62 Strike Price on the option = $15.00 Option life = 103/365 = 0.2822 Standard Deviation in ln(stock prices) = 81% Riskless rate = 4.63% Inputting these numbers into the model, we get Using the normal distribution, we can estimate the N(d1) and N(d2) Since the call is trading at $2.00, it is slightly overvalued, assuming that the estimate of standard deviation used is correct. Implied Volatility The only input on which there can be significant disagreement among investors is the variance. While the variance is often estimated by looking at historical data, the values for options that emerge from using the historical variance can be different from the market prices. For any option, there is some variance at which the estimated value will be equal to the market price. This variance is called an implied variance. Consider the Cisco option valued in the last illustration. With a standard deviation of 81%, we estimated the value of the call option with a strike price of $15 to be $1.87. Since the market price is higher than the calculated value, we tried higher standard deviations and at a standard deviation 85.40%, the value of the option is $2.00. This is the implied standard deviation or implied volatility.. Model Limitations and Fixes The Black-Scholes model was designed to value options that can be exercised only at maturity and on underlying assets that do not pay dividends. In addition, options are valued based upon the assumption that option exercise does not affect the value of the underlying asset. In practice, assets do pay dividends, options sometimes get exercised early and exercising an option can affect the value of the underlying asset. Adjustments exist. While they are not perfect, adjustments provide partial corrections to the BlackScholes model. 1. Dividends The payment of a dividend reduces the stock price; note that on the ex-dividend day, the stock price generally declines. Consequently, call options will become less valuable and put options more valuable as expected dividend payments increase. There are two ways of dealing with dividends in the Black Scholes: * Short-term Options: One approach to dealing with dividends is to estimate the present value of expected dividends that will be paid by the underlying asset during the option life and subtract it from the current value of the asset to use as S in the model. Modified Stock Price = Current Stock Price - Present value of expected dividends during the life of the option * Long Term Options : Since it becomes impractical to estimate the present value of dividends as the option life becomes longer, we would suggest an alternate approach. If the dividend yield (y = dividends/current value of the asset) on the underlying asset is expected to remain unchanged during the life of the option, the Black-Scholes model can be modified to take dividends into account. C = S e-yt N(d1) - K e-rt N(d2) From an intuitive standpoint, the adjustments have two effects. First, the value of the asset is discounted back to the present at the dividend yield to take into account the expected drop in asset value resulting from dividend payments. Second, the interest rate is offset by the dividend yield to reflect the lower carrying cost from holding the asset (in the replicating portfolio). The net effect will be a reduction in the value of calls estimated using this model. stopt.xls: This spreadsheet allows you to estimate the value of a short term option, when the expected dividends during the option life can be estimated. ltopt.xls: This spreadsheet allows you to estimate the value of an option, when the underlying asset has a constant dividend yield. Assume that it is March 6, 2001 and that AT&T is trading at $20.50 a share. Consider a call option on the stock with a strike price of $20, expiring on July 20, 2001. Using past stock prices, the variance in the log of stock prices for AT&T is estimated at 60%. There is one dividend, amounting to $0.15, and it will be paid in 23 days. The riskless rate is 4.63%. In recent years, the CBOT has introduced longer term call and put options on stocks. On AT&T, for instance, you could have purchased a call expiring on January 17, 2003, on March 6, 2001. The stock price for AT&T is $20.50 (as in the previous example). The following is the valuation of a call option with a strike price of $20. Instead of estimating the present value of dividends over the next two years, we will assume that AT&Ts dividend yield will remain 2.51% over this period and that the riskfree rate for a two-year treasury bond is 4.85%. The inputs to the Black-Scholes model are: S = Current asset value = $20.50 K = Strike Price = $20.00 Time to expiration = 1.8333 years Standard Deviation in ln(stock prices) = 60% Riskless rate = 4.85% Dividend Yield = 2.51% The value from the Black Scholes is: The call was trading at $5.80 on March 8, 2001. 2. Early Exercise The Black-Scholes model was designed to value options that can be exercised only at expiration. Options with this characteristic are called European options . In contrast, most options that we encounter in practice can be exercised any time until expiration. These options are called American options . The possibility of early exercise makes American options more valuable than otherwise similar European options; it also makes them more difficult to value. In general, though, with traded options, it is almost always better to sell the option to someone else rather than exercise early, since options have a time premium, i.e., they sell for more than their exercise value. There are two exceptions. One occurs when the underlying asset pays large dividends, thus reducing the expected value of the asset. In this case, call options may be exercised just before an ex-dividend date, if the time premium on the options is less than the expected decline in asset value as a consequence of the dividend payment. The other exception arises when an investor holds both the underlying asset and deep in-the-money puts, i.e., puts with strike prices well above the current price of the underlying asset, on that asset and at a time when interest rates are high. In this case, the time premium on the put may be less than the potential gain from exercising the put early and earning interest on the exercise price. There are two basic ways of dealing with the possibility of early exercise. One is to continue to use the unadjusted Black-Scholes model and regard the resulting value as a floor or conservative estimate of the true value. The other is to try to adjust the value of the option for the possibility of early exercise. There are two approaches for doing so. One uses the Black-Scholes to value the option to each potential exercise date. With options on stocks, this basically requires that we value options to each ex-dividend day and choose the maximum of the estimated call values. The second approach is to use a modified version of the binomial model to consider the possibility of early exercise. In this version, the up and down movements for asset prices in each period can be estimated from the variance and the length of each period2. |
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